Introduction to Noncommutative Algebra

by Matej Bresar

Overview

This textbook offers an elementary introduction to noncommutative rings and algebras. Beginning with the classical theory of finite-dimensional algebras, it then develops a more general structure theory of rings, grounded in modules and tensor products. The final chapters cover free algebras, polynomial identities, and rings of quotients.

Many results are presented in a simplified form rather than in full generality, with an emphasis on clear and understandable exposition. Prerequisites are kept to a minimum, and new concepts are introduced gradually and carefully motivated. Introduction to Noncommutative Algebra is thus accessible to a broad mathematical audience, though it is primarily intended for beginning graduate students and advanced undergraduates encountering the subject for the first time.

This new edition includes several additions and improvements, while preserving the original text’s character and approach.

Praise for the first edition:

“It will soon find its place in classrooms” — Plamen Koshlukov, Mathematical Reviews

“Very well written [...] very pleasant to read” — Veereshwar A. Hiremath, zbMATH

“An excellent choice for a first graduate course” — D. S. Larson, Choice

Product Details

ISBN-13:	9783031962950
Publisher:	Springer Nature Switzerland
Publication date:	09/30/2025
Series:	Universitext
Edition description:	Second Edition 2025
Pages:	234
Product dimensions:	6.10(w) x 9.25(h) x (d)

About the Author

Matej Brešar is a Professor of Mathematics at University of Ljubljana and University of Maribor. His research focus lies in noncommutative algebra and its applications. He is the author or co-author of over 175 research papers as well as four books published by Springer Nature.

Table of Contents

Chapter 1. Finite Dimensional Division Algebras.- Chapter 2. Structure of Finite Dimensional Algebras.- Chapter 3. Modules and Semisimple Rings.- Chapter 4. Structure of Rings.- Chapter 5. Tensor Products in Noncommutative Algebra.- Chapter 6. Noncommutative Polynomials.- Chapter 7. Rings of Quotients and Structure of PI-Rings.

Editorial Reviews

The abundant and carefully chosen examples in the text are selected in such a way as to illustrate the exposition and give some insights into what is going on. ... The book is very good. It will soon find its place in classrooms for most courses in ring theory. I personally liked it very much, and in 2014 our department included the book in the principal bibliography for the corresponding graduate course in noncommutative algebra.” (Plamen Koshlukov, Mathematical Reviews, 2018)

“‘Introduction to noncommutative algebra’ is a very well written book and it is very pleasant to read. … I was very much impressed by the lists of exercises given at the end of each chapter as some of the exercises are not found in standard texts on ring theory.” (Veereshwar A. Hiremath, zbMATH 1334.16001, 2016)

“Students of mathematics typically encounter examples of noncommutative algebraic structures in a first course in abstract algebra, but systematic study of such objects is reserved for a more advanced course. This offering by Brešar (Univ. of Ljubljana, Slovenia) is written in a style governed by two guiding principles. … an excellent choice for a first graduate course in noncommutative algebra. Summing Up: Highly recommended. Graduate students and researchers/faculty.” (D. S. Larson, Choice, Vol. 52 (10), June, 2015)

From the Publisher